Crossing Numbers and Stress of Random Graphs

TL;DR

This paper investigates the crossing numbers and stress of random geometric graphs, demonstrating that planar projections approximate crossing numbers within a constant factor and that crossing number correlates with stress.

Contribution

It introduces bounds on crossing numbers for random geometric graphs and establishes their positive correlation with stress in projections.

Findings

Projection approximates crossing number within a constant factor

Crossing number is positively correlated with stress

Provides theoretical bounds for random geometric graphs

Abstract

Consider a random geometric graph over a random point process in $\mathbb{R}^d$. Two points are connected by an edge if and only if their distance is bounded by a prescribed distance parameter. We show that projecting the graph onto a two dimensional plane is expected to yield a constant-factor crossing number (and rectilinear crossing number) approximation. We also show that the crossing number is positively correlated to the stress of the graph's projection.